reserve T,U for non empty TopSpace;
reserve t for Point of T;
reserve n for Nat;

theorem Th8:
  for T being trivial non empty TopSpace
  for t being Point of T, L being Loop of t holds
  the carrier of pi_1(T,t) = { Class(EqRel(T,t),L) }
  proof
    let T be trivial non empty TopSpace;
    let t be Point of T;
    set E = EqRel(T,t);
    let L be Loop of t;
    thus the carrier of pi_1(T,t) c= { Class(E,L) }
    proof
      let x be object;
      assume x in the carrier of pi_1(T,t);
      then consider P being Loop of t such that
A1:   x = Class(E,P) by TOPALG_1:47;
      P = I[01] --> t by TOPREALC:22
      .= L by TOPREALC:22;
      hence x in { Class(E,L) } by A1,TARSKI:def 1;
    end;
    let x be object;
    assume x in { Class(E,L) };
    then
A2: x = Class(E,L) by TARSKI:def 1;
    L in Loops t by TOPALG_1:def 1;
    then x in Class E by A2,EQREL_1:def 3;
    hence thesis by TOPALG_1:def 5;
  end;
